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INTRODUCTION TO THE THEORY OF QUANTUM MEASUREMENTS
(M.B.Mensky)

Contents of the course

      In recent decades, the theory of quantum measurements has received a new development and has led to the emergence of concepts such as decoherence and quantum nonlocality, which in turn gave rise to new applications of quantum mechanics. Phenomena such as quantum measurements occur in the interaction of quantum systems (for example, complexes of molecules) and the environment and significantly modify their dynamics. This allows, in particular, to construct a quantum theory of open systems and in particular the quantum theory of dissipation..
      Quantum mechanics differs from classical by the fact that states of quantum systems form a linear space (i.e. the superposition of states is possible) and their change over time is described by the linear Schr?dinger equation. These purely mathematical features of quantum systems lead to profound physical differences between quantum systems and classical. One major difference is the different measurements. Predictions of the result of measurement of quantum systems are probabilistic in nature, even if its condition before the measurement is known exactly. In addition, the measurement system will inevitably change under the influence of measurement, which is a «pay» for the information received.
      Specific features of quantum measurement described by von Neumann postulate, that when measuring a quantum system subjected to the reduction of its state (in other terms - the wave function collapse occurs). A deeper analysis shows that when measuring the entanglement of the measured system comes with a measuring device (in the more general case - the environment). In another terminology, a quantum correlation is established between them. In this case the system is exposed to decoherence. Physically, this means that the system partially loses its specifically quantum features, «becomes classical to some extent». Role of decoherence was fully understood only in the 80's of last century. Decoherence process occurs not only in the measurement of the system (to get information about it), but when uncontrolled interaction of the system with its surroundings happens. This explains, in particular, how quantum dissipation of the system occurs.
      Possibility of quantum correlation (entanglement) between a quantum system gives rise to specific phenomena, called quantum nonlocalness. The phenomenon of quantum non-locality is illustrated by the situation described as far back as 1935 in the famous work of Einstein-Podolsky-Rosen (EPR). Variants of the situation are the EPR are spin correlation of two electrons or correlated polarizations of two photons. In this case, the measurement of the spin projection of one of the electrons (or the polarization of a photon) determines the spin projection of the second electron (respectively - the second photon polarization), despite the lack of interaction between them.
      In 1964, John Bell brought up the inequality for the probabilities of various measurements in the EPR situation, which must be met if the observable (projection of spin of electrons or polarization of photons) are real in the sense in which reality is understood in classical physics. Later in the Aspec's experiments it was shown that the Bell inequality is violated. Thus, it was experimentally proved that the classic understanding of reality is not applicable in quantum mechanics.
      In radical contrast to the classical quantum reality based, new applications of quantum mechanics, which have been substantiated and partially implemented in recent decades as part of a new scientific field - quantum information science. Quantum cryptography allows you to send secret messages so that a secret may not be affected (it is protected by the fundamental laws of quantum mechanics). The procedure for quantum teleportation allows to move the state in remote locations, i.e. bring the system located in a remote area in the state where the system is localized in the near field. Finally, the quantum computers of sufficient power at the expense of quantum parallelism could carry out such calculations for a reasonable time, for which time of Universe living is not enough for classical computers.

Plan of the course

  • • Fundamental principles of quantum mechanics
    • The space of states and operators of observables
    • Hamiltonian and the Schrodinger equation
    • Harmonic oscillator
  • The specificity of measurement in quantum mechanics
    • The opposite effect of measurement on the state of the system as payment for information
    • Principle of uncertainty: the impact of the device or the intrinsic properties of the system?
    • The lack of determinism and absolute probabilities
  • The formalism of quantum measurement
    • Reduction of the state during the measurement (the collapse of the wave function)
    • Entanglement of states of measured system and its environment
  • Decoherence of the system when measuring
    • Density matrix formalism
    • Induced density matrix of the measured system
    • Decoherence of the measured system
    • Dissipation of quantum system
  • Quantum nonlocality
    • EPR type experiment and quantum correlation
    • RCS Experience the type and quantum correlation
    • Experiments of Aspec and denial of reality of local observables
  • Fundamentals of quantum informatics
    • Quantum cryptography
    • Quantum teleportation
    • Quantum parallelism and quantum computer

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